Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I know that for a compact manifold $M$ any maximal ideal in the algebra $C^\infty(M)$ of smooth functions on $M$ is of the form $\mathfrak{m}_p=\{f\in C^\infty(M)|f(p)=0\}$. For example, the proof is in the answer to this question.

Is that true for non-compact manifolds? For example, for $M=\mathbb{R}$? I don't think so, but I can't figure out a counterexample. Do you know any example of a maximal ideal in $C^\infty(\mathbb{R})$ which is not of the form $\mathfrak{m}_p$?

Thank you very much!

Edit: Thank you very much for pointing out the maximal ideal, containing the ideal of compactly-supported functions. Does anyone know how to describe it?

share|improve this question

2 Answers 2

up vote 4 down vote accepted

I don't have an explicit example at hand, but there are other ideals than $\mathfrak m_p$. Note that $C_0^\infty$, the set of compactly supported smooth functions is an ideal, not contained in any $\mathfrak m_p$. By Zorn's lemma, every ideal is contained in some maximal ideal, so there must be others.

share|improve this answer
    
Yes, that's nice! This example didn't come to my mind. –  Sasha Patotski Dec 17 '13 at 1:21

No. For example, functions with compact support form an ideal.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.